THE LESSER NAMES – THE TEACHERS OF THE EDINBURGH MATHEMATICAL SOCIETY AND OTHER ASPECTS OF SCOTTISH MATHEMATICS, 1867-1946 Marit Hartveit A Thesis Submitted for the Degree of PhD at the University of St. Andrews 2011 Full metadata for this item is available in Research@StAndrews:FullText at: https://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/1700 This item is protected by original copyright The Lesser Names — The teachers of the Edinburgh Mathematical Society and other aspects of Scottish mathematics, 1867–1946 Marit Hartveit AThesissubmittedfortheDegreeofPhD at the School of Mathematics and Statistics The University of St Andrews 2010 I, Marit Hartveit, hereby certify that this thesis, which is approximately 61000 words in length, has been written by me, that it is the record of work carried out by me and that it has not been submitted in any previous application for a higher degree. I was admitted as a research student in February 2007 and as a candidate for the degree of PhD in February 2007; the higher study for which this is a record was carried out in the University of St Andrews between 2007 and 2010. Date …… signature of candidate ……… I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of PhD in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. Date …… signature of supervisor ……… In submitting this thesis to the University of St Andrews I understand that I am giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. I also understand that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona fide library or research worker, that my thesis will be electronically accessible for personal or research use unless exempt by award of an embargo as requested below, and that the library has the right to migrate my thesis into new electronic forms as required to ensure continued access to the thesis. I have obtained any third-party copyright permissions that may be required in order to allow such access and migration, or have requested the appropriate embargo below. The following is an agreed request by candidate and supervisor regarding the electronic publication of this thesis: Access to printed copy and electronic publication of thesis through the University of St Andrews. Date …… signature of candidate …… signature of supervisor ……… ii Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv A note on the notation for sessions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 1 The Early Days of the EMS 1 1.1 The Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Society’s activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Election . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Subscription fees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Members by occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 The Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 The Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.1 The Proceedings of the Edinburgh Mathematical Society 15 1.5.2 The unread papers and the unpublished papers . . . . . . . . . . . . . . . . . . . 16 1.5.3 Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.4 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.5 The Mathematical Notes 22 1.5.6 Exchanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.7 Financing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 The Mathematics of the Teachers 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 The Papers in the Proceedings 27 2.2.1 The numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.2 The topics of the teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.3 The types of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 The papers of John Watt Butters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1 ‘On the solution of the equation x p − 1=0(p being a prime number)’ . . . . . . 37 2.3.2 ‘Notes on factoring’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3.3 ‘A geometrical proof of certain trigonometrical formulae’ . . . . . . . . . . . . . . 47 2.3.4 ‘Elementary notes’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3.5 ‘Notes on decimal coinage and approximation’ . . . . . . . . . . . . . . . . . . . 57 2.3.6 ‘On the decimalization of money’ . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.7 ‘On the use of symmetry in geometry’ . . . . . . . . . . . . . . . . . . . . . . . . 64 iii CONTENTS CONTENTS 2.3.8 Butters’s publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3 The Enumeration of Rhyme Schemes 77 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.1.1 Bell numbers and Stirling numbers of the second kind . . . . . . . . . . . . . . . 77 3.2 Aitken’s article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2.1 Introduction: Grouping individuals . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2.2 Partitions of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.3 The generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.4 Repeated differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2.5 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2.6 Dobi´nski’s result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2.7 Differences of zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.2.8 Recurrence relation and Aitken’s Array . . . . . . . . . . . . . . . . . . . . . . . 86 3.2.9 Prime number division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.2.10 Asymptotic expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.3 Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.3.1 Aitken to Thompson, 6 Dec. 1938 . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.3.2 Aitken to Thompson, 19 Dec. 1938 . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.3.3 Thompson to Bennett, 20 Dec. 1938 . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.3.4 Bennett to Thompson, 27 Dec. 1938 . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3.5 Bennett to Thompson, postcard, 29 Dec. 1938 . . . . . . . . . . . . . . . . . . . 96 3.3.6 Bennett to Thompson, 30 Dec. 1938 . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3.7 Bennett to Thompson, 30 Dec. 1938 . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.3.8 Bennett to Thompson, 1 Jan. 1939 . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3.9 Aitken to Thompson, 4 Jan. 1939 . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.3.10 Bennett to Thompson, 18 Jan. 1939 . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.3.11 Thompson to Bennett, 19 Jan. 1939 . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.3.12 Bennett to Thompson, 27 Jan. 1939 . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.3.13 Thompson to Bennett, 1 February 1939 . . . . . . . . . . . . . . . . . . . . . . . 107 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4 The Road to Research 111 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1.1 The outset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1.2 The key players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2 A new policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.1 The first discussions, 1926–28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.2 Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.3 The teachers in 1926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.2.4 The discussions at the committee meetin gs . . . . . . . . . . . . . . . . . . . . . 119 4.2.5 A new journal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3 The Controversy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3.1 The Second Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3.2 MacRobert’s resignation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.3.3 A new course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.3.4 The Glasgow Lapse and aftermath . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.3.5 The reasons for the controversy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.3.6 Further developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 iv CONTENTS CONTENTS 4.3.7 A new generation of teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5 The Higher Education of Women 137 5.1 Women in the early society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.2 How Flora got her Cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.2.1 Flora Philip’s love of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.2.2 The mathematical classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Appendix A: Tables 157 Appendix B: The rhyme scheme letters 169 Appendix C: The EMS letters 195 v CONTENTS CONTENTS vi List of Figures 1.1 Members by occupation (1883–1946) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Members by occupation — by % (1883–1946) . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Committee members by occupati on — by % (1883–1946) . . . . . . . . . . . . . . . . . 13 1.4 Teachers and academics in office . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Teachers and academics on the committee . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Talks unpublished and papers unread . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.7 Authors in Proceedings 1883–1887 and 1918–1922 . . . . . . . . . . . . . . . . . . . . . . 18 1.8 Authors in Proceedings 1923–1926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.9 Papers by subject, Proceedings Series 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.10 Authors in the Notes 1909–1946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1 Papers by teachers and academics, Proceedings Vol. 1–44 . . . . . . . . . . . . . . . . . . 28 2.2 Papers per member, teachers and academics, Proceedings Vol. 1–44 . . . . . . . . . . . . 28 2.3 Papers by teachers and academics, PEMS and Notes 29 2.4 Papers by teachers according to subject, Vol. 1–44 . . . . . . . . . . . . . . . . . . . . . 30 2.5 Papers by teachers according to subject, 5-year intervals. . . . . . . . . . . . . . . . . . 30 2.6 Papers by academics according to subject, Vol. 1–44 . . . . . . . . . . . . . . . . . . . . 32 2.7 Papers by teachers according to type, Vol. 1–44 . . . . . . . . . . . . . . . . . . . . . . . 34 2.8 Papers by teachers according to type, 5-year intervals . . . . . . . . . . . . . . . . . . . 34 2.9 John Watt Butters (1863–1946) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.10 Projections on the axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.11 The general formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.12 Sums of cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.13 Wrong use of the term ‘produced’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.14 Proposition I.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.15 Proposition I.27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.16 Propositions I.9–10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.17 Areas in skew symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.18 Proposition I.36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.19 Proposition I.43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A. C. Aitken . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 D. W. Thompson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 G. T. Bennett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 P. A. MacM ahon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 E. T. Whittaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.1 Thomas Murray MacRobert (1884-1962) . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 vii LIST OF FIGURES LIST OF FIGURES T. M. MacRobert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.1 Women in the Society, 1883–1946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.2 The Edinburgh Mathematical Col l oquium 1913 . . . . . . . . . . . . . . . . . . . . . . . 140 5.3 The Edinburgh Mathematical Col l oquium 1926 . . . . . . . . . . . . . . . . . . . . . . . 141 viii [...]... thorough study The occupations of the members will be placed under particular scrutiny The chapter also considers the organisation of the Society, the Committee and its office-bearers, again focussing on the occupations of the people involved Most of the remainder of the chapter is then devoted to a detailed study of the Society s two periodicals The Proceedings of the Edinburgh Mathematical Society developed... — The Edinburgh Mathematical Society • EMS Archives — Archive of the Edinburgh Mathematical Society • LMS — The London Mathematical Society • Notes — The The Edinburgh Mathematical Notes • OHE — Other forms of higher education, such as teacher training and technical training outside universities • PEMS — The Proceedings of the Edinburgh Mathematical Society • PLMS — The Proceedings of the London Mathematical... within the Committee, with the resignation of two of its members, the cessation of the annual Glasgow meetings, and a complete revision of the programme for the following year The chapter aims to answer two questions What made the Society turn towards research, and did this mean there could not be a place for the teachers anymore? Chapter 5 regards the women in the earliest days of the Society There... standard of mathematics, but even more to a change in the Society s printing policy in the 1920s, that turned the Society s Proceedings into a pure research publication and led to the death of the ‘teacher journal’, the Mathematical Notes The thesis also argues that this change, drastic as it may seem, does not represent a change in the Society s nature For this aim, the role of the teachers within the. .. Bridge At the behest of the Astronomer Royal of Scotland, there was also an excursion to the Observatory on the 29th of March 1901 The Colloquia The well-known St Andrews colloquia of the Society began in 1926, but before that, the Society held two colloquia in Edinburgh in 1913 and 1914.9 The first one was the first of its kind in Britain and was set up largely to explore Professor Whittaker’s new Mathematical... Chapter 1 provides the necessary background for the other chapters, as well as presenting aspects of the Society s history that have not been examined before The chapter describes the circumstances regarding the foundation of the Society, such as who the founders were, and discusses why the Society was formed The various activities, mainly the meetings, will be explored, before the membership and its development... LIST OF TABLES LIST OF TABLES x ABSTRACT Abstract The Edinburgh Mathematical Society started out in 1883 as a society with a large proportion of teachers Today, the member base is mainly academical and there are only a few teachers left This thesis explores how and when this change came about, and discusses what this meant for the Society It argues that the exit of the teachers is related to the rising... Mathematical Society • StASC — The Special Collections at the University Library of the University of St Andrews xviii Chapter 1 The Early Days of the EMS 1.1 The Foundation The Edinburgh Mathematical Society was founded on the 2nd of February 1883, when 53 gentlemen met in the Mathematical Classroom at Edinburgh University They were there because they had all received a certain circular, proposing the establishment... with the events leading to the migration of the EMS into a research society and the exit of the teachers, focussing largely on a debate on the publications that took place between 1926 and 1931 The scene will be set, as it were, with a summary of the situation when this discussion began The path towards the debate’s culmination in 1931 is traced out; then an explanation of the controversy itself and the. .. required for this thesis The aforementioned digitalisation of the minute books consisted of summarising every meeting held in the first 64 sessions, from 1883 up to the end of the Second World War The sum2 See [64] and [63] xv INTRODUCTION maries contain the location of the meeting, the name of the chairman, the papers that were read and any members that were elected If anything out of the ordinary took